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Setting Let $(M,g)$ be a compact Riemannian manifold with smooth boundary, and let $\nabla$ be its associated Levi-Civita connection. Consider the following formally self-adjoint, second order linear ...

Let $M$ be a compact manifold equipped with finite rank vector bundles $E$ and $F$ with spaces of $C^{\infty}$ sections denoted $\Gamma(E)$ and $\Gamma(F)$ respectively. It is standard that a ...

A major open problem in submanifold geometry is to determine whether there exists a nontotally geodesic isometric immersion of $\mathbb{S}^n$ into $\mathbb{S}^{2n}$. Every isometric immersion of $\...

Asked this on Math Stack Exchange awhile ago but it got ignored then deleted. To solve a differential equation of one variable, you need constraints equal to the number of derivatives. For a partial ...

I hope to extend the PDEs across the boundary using Cauchy–Kowalevski Theorem. Given a real analytic manifold $M$ with boundary $\partial M$, the solution $u$ to the PDE $$\triangle u+b(x)\nabla u+c(...